General conditions for global stability in a single species population-toxicant model

Abstract

We deal with the global stability for a well-known population-toxicant model. We make use of a geometrical approach to the global stability analysis for ordinary differential equation which is based on the use of a higher-order generalization of the Bendixson's criterion. We obtain sufficient conditions for the global stability of the unique nontrivial equilibrium. These conditions are expressed in terms of a generic functional describing the population dynamics. In the special case of a logistic-like population dynamics, we get conditions which improve the ones previously known, obtained by means of the Lyapunov direct method.

Introduction

Let T(t), U(t) and N(t) represent at time t the concentration of toxicant in the environment, the concentration of the toxicant in the total population and the population biomass, respectively. Mass balance arguments drive to the following model:

$$\text{T}\text{̇}\text{=Q−δ}{}_0\text{T−αNT+πγNU,}\text{U}\text{̇}\text{=αNT−δ}{}_1\text{U−γNU,}\text{N}\text{̇}\text{=}\text{F}\text{(T,U,N),}$$

where $\text{F}$ is a sufficiently smooth functional such that the initial value problem given by (1) and T(0)⩾0, U(0)=kN(0), N(0)⩾0 admits a unique and continuable solution for all positive time.

This model, when the functional $\text{F}$ is given by

$$\text{F}\text{(T,U,N)=r(U)N−}\text{r}{}_0\text{K(T)}\text{N}{}^2$$

is due to Freedman and Shukla [8].

The roots of model (1) in the biomathematics literature are quite deep. In the eighties a deterministic approach to the problem of developing mathematical modelling to assess the effects of pollutants on ecological system was proposed by Hallam and his coworkers [11], [12]. Their model is nowadays classic. It takes into account of three state variables (expressed as biomass or concentrations): a population, a toxicant in the environment (external toxicant) and the toxicant in the total population (internal toxicant). The toxicants dynamics are founded on mass balance equation. The population dynamics is assumed to be logistic whereas the coupling between the population and the toxicant is accomplished by a linear dose–response function, that is the population growth rate is assumed to be a linear function of the internal toxicant.

The key modelling feature of this model stands in the equations ruling the toxicant dynamics. In fact, the same toxicant dynamics has been proposed with different population equations (see e.g. [11], [12], [23]) and has been coupled to single and multi species problems in several contexts as Lotka Volterra and chemostat-like environment [5], [18].

Ordinary, delay-differential and stochastic models based on the Hallam's modelling idea can be found in the literature (e.g. [4], [6], [9]) and the ordinary basic model itself, or its slightly modified versions, is still subject of investigations (see e.g. [23]).

In the Hallam's model, the internal toxicant is defined with respect to the biomass of the total population, so that the internal toxicant dynamics does not explicitly depend on the density of the population.

In 1991 Freedman and Shukla [8] proposed a different version, model , , which is based on the consideration that the internal toxicant should be defined with respect to the total mass or volume of the environment where the population lives. This consideration, which makes the model of the ecotoxicological problem more visible, has been taken into account also in [1], where an extention to the spatial structured case is presented, resulting in a diffusive–convective model. A diffusion model is also considered in [2].

The papers dealing with these models usually provide results on the qualitative properties of solutions. In this direction, the stability analysis plays an important role. The nature of equilibria may be interpreted in terms of population survival or extinction, though different approaches based on the concepts of persistence and permanence may also give an account on the population fate (see e.g. [19]).

In this paper, we deal with the global stability for the population-toxicant model proposed by Freedman and Shukla. We allow the population dynamics to be ruled by a generic functional $\text{F}$ and find conditions that such functional have to satisfy in order to guarantee the global stability of the unique nontrivial equilibrium. As a particular case, we consider a global stability problem solved in [8] by means of the Lyapunov direct method and improve the conditions there obtained.

We make use of a geometric approach to the global stability analysis for ordinary differential equations emerged from a series of papers in the middle of 1990s (see [17] and the references contained therein) and recently applied to several epidemic models [3], [14], [16], [24].

Such method is based on the use of a higher-order generalization of the Bendixson's criterion. The existence of a robust Bendixson criterion for the system at hand, associated to the existence and unicity of an equilibrium, determines the global dynamics of the system. The main theorem of the method requires the use of a Lozinskiı̆ logarithmic norm.

The paper is organized as follows: in Section 2 the mathematical framework for proving the global stability is outlined. In Section 3, the population-toxicant model is introduced and sufficient conditions for the global stability of the unique nontrivial equilibrium are obtained in terms of the functional describing the population dynamics. In Section 4, the result is specialized for the logistic-like population dynamics proposed by Freedman and Shukla. Numerical verifications, useful to better understand the relationships between the different parameter restrictions are presented in Section 5.